Problem: Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle MON = 4x + 22$, and $ m \angle LOM = 5x + 23$, find $m\angle MON$. $O$ $L$ $N$ $M$
Solution: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {5x + 23} + {4x + 22} = {90}$ Combine like terms: $ 9x + 45 = 90$ Subtract $45$ from both sides: $ 9x = 45$ Divide both sides by $9$ to find $x$ $ x = 5$ Substitute $5$ for $x$ in the expression that was given for $m\angle MON$ $ m\angle MON = 4({5}) + 22$ Simplify: $ {m\angle MON = 20 + 22}$ So ${m\angle MON = 42}$.